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<h1>Euler Angles</h1>


<p>The <a href="http://en.wikipedia.org/wiki/Euler_angles">Euler angles</a> are three angles that describe the orientation of a rigid body. There are twelve different conventions when talking about Euler angles. They can be grouped into <strong>Proper Euler angles</strong>, and <strong>Tait-Bryan angles</strong> where the latter represents rotations about three distinct axes. In CoppeliaSim, we use <strong>Tait-Bryan angles</strong> <em>alpha</em>, <em>beta</em> and <em>gamma</em> (or (a,b,g)) describing a rotation composed by three elemental rotations:</p>

<p align=center><img src="images/eulerAngles.jpg"></p>
<p class=imageLabel>[Euler angles convention in CoppeliaSim]</p>
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<p>where Rz, Ry and Rx represent elemental rotations about axes Z, Y and X respectively (in that order), of the <strong>absolute</strong> reference frame.</p>
<p>For a rigid body, or <a href="objects.htm">scene object</a>, you can obtain the same transformation if you rotate it in following order about its <strong>own</strong> reference frame: rotation of <em>alpha</em> about own x, followed by a rotation of <em>beta</em> about own y, followed by a rotation of <em>gamma</em> about own z.</p>
<p><strong>Yaw-Pitch-Roll</strong> are also <strong>Tait-Bryan angles</strong> describing a rotation composed by three elemental rotations:</p>

<p align=center><img src="images/yawPitchRollAngles.jpg"></p>
<p class=imageLabel>[Yaw-Pitch-Roll rotations]</p>
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<p>where Rx, Ry and Rz represent elemental rotations about axes X, Y and Z respectively (in that order), of the <strong>absolute</strong> reference frame.</p>

<p>To convert between <em>alpha-beta-gamma</em> angles and <em>yaw-pitch-roll</em> angles, you can use <a href="regularApi/simYawPitchRollToAlphaBetaGamma.htm">sim.yawPitchRollToAlphaBetaGamma</a> or <a href="regularApi/simAlphaBetaGammaToYawPitchRoll.htm">sim.alphaBetaGammaToYawPitchRoll</a> functions.</p>


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